The popularity of this interpretation is certainly due to the Haggada
צֵא וּלְמַד מַה בִּקֵּשׁ לָבָן הָאֲרַמִּי לַעֲשׂוֹת לְיַעֲקֹב אָבִינוּ: שֶׁפַּרְעֹה לֹא גָזַר אֶלָּא עַל הַזְּכָרִים, וְלָבָן בִּקֵּשׁ לַעֲקֹר אֶת־הַכֹּל. שֶׁנֶּאֱמַר: אֲרַמִּי אֹבֵד אָבִי, וַיֵּרֶד מִצְרַיְמָה וַיָּגָר שָׁם בִּמְתֵי מְעָט, וַיְהִי שָׁם לְגוֹי גָּדוֹל, עָצוּם וָרָב.
Go out and learn what Lavan the Aramean sought to do to Ya'akov, our father; since Pharaoh only decreed [the death sentence] on the males but Lavan sought to uproot the whole [people]. As it is stated (Deuteronomy 26:5), "An Aramean was destroying my father and he went down to Egypt, and he resided there with a small number and he became there a nation, great, powerful and numerous."
Why the Haggada assumed this interpretation can probably be found in some of the 50 commentaries found in the link.
But I think the violent threat of Lavan is certainly implicated in parshas Vayeitzei.
See 31:25 where Hashem Himself had to appear to Lavan and warn him not to engage Yaakov for good or evil:
וַיָּבֹ֧א אֱלֹהִ֛ים אֶל־לָבָ֥ן הָאֲרַמִּ֖י בַּחֲלֹ֣ם הַלָּ֑יְלָה וַיֹּ֣אמֶר ל֗וֹ הִשָּׁ֧מֶר לְךָ֛ פֶּן־תְּדַבֵּ֥ר עִֽם־יַעֲקֹ֖ב מִטּ֥וֹב עַד־רָֽע׃
But God appeared to Laban the Aramean in a dream by night and said to him, “Beware of attempting anything with Jacob, good or bad.”
Then, when Lavan meets up with Yaakov he offers a veiled threat:
יֶשׁ־לְאֵ֣ל יָדִ֔י לַעֲשׂ֥וֹת עִמָּכֶ֖ם רָ֑ע וֵֽאלֹהֵ֨י אֲבִיכֶ֜ם אֶ֣מֶשׁ ׀ אָמַ֧ר אֵלַ֣י לֵאמֹ֗ר הִשָּׁ֧מֶר לְךָ֛ מִדַּבֵּ֥ר עִֽם־יַעֲקֹ֖ב מִטּ֥וֹב עַד־רָֽע׃
I have it in my power to do you harm; but the God of your father said to me last night, ‘Beware of attempting anything with Jacob, good or bad.’
Do not think for a moment Lavan would not have carried through with this evil violence if not for Hashem's intervention.